Let me stipulate up front that my understanding of both quantum physics and general physics is minimal — no more than a typical lay person who has read a few books and papers and has a basic understanding of math only up to the calculus level. So forgive me (and correct me) if my terms are technically unprecise.
As I understand it, the quantum world is essentially unpredictable. The more you know about one attribute of a particle (its vector), the less you know about another attribute (its scalar). The quantum world is a world of possibilities and probabilities. Particles obey laws that are strange in the macro world. Meanwhile, the macro world is fairly predictable. If you know the location and velocity of an asteroid, you can predict with a reasonable certainty where it has been and where it is going.
I guess I can put my question this way: Does the location and velocity of the asteroid match the most probable locations and velocities of its component particles as some sort of sum or average of those? In other words, do the quantum probabilities converge into what we perceive as a moving asteroid?
Yes, you’re quite right - the correspondence principle for quantum mechanics is statistical by nature. Yes, there is a probability of an asteroid “going all quantum” on us and appearing somewhere noticeably different from where it was an instant before. (Indeed, some of its electrons do.) But the probabilities are such that for the whole asteroid (or a noticeable proportion of the electrons/quarks/whatever) to appear a distance much greater than the width of their ground state orbitals away is such that continual observation for the entire lifetime of the universe will still yield a negligible probability.
The weirdness of QM comes from the apparent fact that before you interact with the system in some way, things travel in “probability waves”, and your interaction then reveals the object’s classical history. The asteroid’s probability wave is indistinguishable from its classical counterpart. (In fact, even atoms are so quantumly “big” that the same is largely true of them also.) You have often characterised atoms as “just a probability distribution” - this is accurate, I guess, but only of atoms in the future. Once you have observed them, they are probability distributions no longer: they really are where they are observed to be. In the same way, a dice could be said to be “just a probability distribution” but only until you throw it.
Thanks for your answer, Sentient. For the record, it is the universe that I have called a probability distribution. Atoms I have said are not real. (Though I suppose the descriptions are interchangeable.)
Regarding dice, isn’t it true that they are predictable in theory since they are macro-scale objects? That is, if you knew everything about them, including all exact variables such as wind resistance, acceleration, initial properties, and so forth, wouldn’t you be able to say with a reasonable certainty how they will land?
Well, in that case your observations of your particular region of spacetime are making that region not a probability distribution any more, but since the overwhelming majority of spacetime remains unobserved, I guess your version statistically tends towards the truth.
Yes, they are too big to yield a non-negligible probability of observing quantummy behaviour in them in toto. But in their case, another complication arises (at least, in most practical rolls of a balanced die on a real frictional surface): chaos. Those “exact variables” you need are themselves open to tiny fluctuations in their later decimal places which can ultimately be the straw which breaks the dice-thrower’s luck, and one possible source of these tiny inaccuracies are the quantummy behaviour of the fundamental particles which provide the repulsive force between the dice and table at every bounce.
So, as usual, it’s easier if you first realise that it’s always more complicated than you first realise.
I don’t think you were unclear. What I’m saying is there is no such thing as an exact description of, say, the weight of a die. Any such description would have a finite number of digits, and could always be extended further and more accurately. No?
It’s worse than that. Even if you could measure quantities with infinite precision, you still can’t get all the information you might want. Your position is what leads to the standard misconception of Heisenberg’s uncertainty principle – not only can’t we measure all classical variables exactly, real (i.e. quantum) systems don’t even have all the information.
To pick a simple system to illustrate, consider a particle travelling on a one-dimensional line. Classically, we describe it by its position x and momentum p, where x and p are any real numbers. The state is then just a point in the plane, and the misconception is that we just can’t pin down this point exactly.
“Really”, though, the state is some complex function psi(x) defined on positions. The probability that we see the point between x and x+dx is |psi(x)|[sup]2[/sup]dx, and the integral of this quantity over the line is 1. It turns out we can also describe it by phi§, which is the “Fourier transform” of psi(x)[1]. The probability of finding the momentum of the particle between p and p+dp is |phi§|[sup]2[/sup]dp, and again the integral of this over all p is 1.
Now, to each psi (or phi) – which is just a function on the line – we can find the mean value and the standard deviation of the probability distribution |psi(x)|[sup]2[/sup]dx just as in basic calculus. Call the deviation we get from a state psi(x) Dx (really a capital delta rather than D) and the one we get from the corresponding phi§ Dp. Then it turns out that the product of these quantities, DxDp, is greater than a specific constant. That is, a state psi(x) doesn’t give exact values for either x or p, but only smeared-out values, and the amount of smearing is definitely not zero. That is, it’s not just that we can’t measure the exact values of position and momentum for a state – the state doesn’t even have exact values.
[1] The details are a bit messy, and I’ll leave them out unless specifically requested to explain.
Particles have many properties, some of which are vectors and some of which are scalars. There’s no property of a particle you can point to and say “that’s its vector”. I think what you meant to say is that the more you know about a particle’s momentum, the less you know about its position, but both of these are vectors. There are, incidentally, a great number of pairs of conjugate quantities like this, which cannot both be known at once; position and momentum are just the best-known.
A clarification here, but an important one: the more precise your measurement of the momentum of a particle, the less precise your measurement of the position, and vice versa per the indeterminacy principle. You can apply this relation to other co-related factors–say, velocity and probability density, et cetera, but both are general vector (or perhaps tensor) quantities, not scalars. (Scalars are measurements that do not have an orientation or direction.)
If what you mean by “real” is “made of little bits of stuff” then no; they’re no more like a basketball, say, than would be a sphere of cotton candy. Our image of an atom as a nucleus with electrons spinning around in “orbitals” is just illustrative, and often misleadingly so. But they are very much “real” in the sense that they make up and interact with everything else in the universe. You might as well argue that waves in the ocean aren’t “real” because they are only distrurbances in the water; while it’s true they wouldn’t exist if you took the water away, this doesn’t prevent them from really pounding you as you go through the surf zone, or making you seasick as you rock at anchor.
Even dismissing effects at the quatum level (which may or may not play any significant part in determining the outcome of an event), there is a level of indeterminacy even in the classical level. For instance, although the interactions of celestial bodies via gravitational attraction (and assuming only Newtonian laws, ignoring general relativity) follow exactly deterministic rules, the result is not generally analytically soluable for systems of more than two bodies, even knowing all values (mass, velocity) for a closed system. See the n-body problem. You can determine a solution by iteration, but the quality of your solution depends upon how small your iteration period is; for some solutions, even a subtle change in iteration will result in a bifurcation that will have significant ramifications down the line. The occurance and sensitivity of these sorts of perturbations, even in very simple non-linear but fundamentally deterministic (classical) systems is the basis for chaos theory. (I’m not really happy with the Wiki article in that link–it’s a little misleading or incomplete, even as a brief survey–but it’ll give you the idea of how classical systems can be nonpredictable.)
To further extend the already excellent answers to the final question posed in the OP: an asteroid (or any other body) represents the sum total of information that is accessible to an observer about that body, which is less than all of the information that is “stored” in (or makes up) that body. Note that this depends as much on the observer as it does on the body being observed; there is a fundamental limit–analogous to thermodynamic entropy on the stochastic level–regarding how much information you can extract about the nature and condition of any collection of particles. This uncertainty increases as the system becomes more massive, owing to the interrelations of the components of the body, until, as SentientMeat has already noted, the system of quantum particles becomes so large that it can be treated classically as a single lump of material as defined by the correspondence principle. This doesn’t mean that quantum effects can’t be exhibited on a “macro” scale, but that they only do so in a statistical fashion, i.e. the percentage likelyhood of electrons to “tunnel” through energy boundries. Exceptions to this can be found in supersolids and superfluids like a Bose-Einstein condensate but they only appear at the extremes of environment.
So from any real world observation–down to the molecular level–we “see” or percieve atoms and their components as behaving like “real” particles, albeit some that interact occasionally in strange ways, but nonetheless can be modeled as deterministic (classical mechanics) or stochastic (thermodynamics, statistical mechanics). If we want to understand the nature of the rules regarding the interactions of individual particles, then we have to delve into the arcane and nonintuititive field of quantum mechanics. I don’t think, strictly speaking, that it is appropriate to say that the probabilities “converge” to our perception of the asteroid, but you could say that the asteroid represents an average of the probability density of each particle about it’s loci to the point that the system is indistinguishable from a solid mechanical continuum.
Well, no, that’s not what I mean, but if I explained it to you, it would take us into philosophical issues that might be inappropriate for a mere factual forum.
Is one macro object distinguished from another by electromagnetic force? In other words, is the essential difference between my feet and the floor under them the fact that the forces binding together the atoms in my feet do not interact with the forces that are binding together the atoms in the floor? Why don’t my feet converge with the floor and become one object whenever the outermost layer of atoms for each come into contact?
You mean that if we tried to measure some better way we’d get the information? Rubbish. Pay attention this time.
The argument I gave proving the UP has nothing to do with how the measurements are performed. It isn’t at all a question of being able to squeeze a little bit more accuracy out, but that states (in QM) fundamentally do not have that information in them. If you have an exact position, the Fourier transform is evenly spread-out and there is no information whatsoever about the momentum. Every other kind of position state lacks exact information about position, and the amount of precision in the momentum information is inherently limited.
Like it or not, the property states inherent in QM form a highly non-Boolean lattice. Classical logic falls apart and is replaced by something else entirely, the distinction being governed by functional analysis and measured by Planck’s constant. The miracle is that when Planck goes to zero (as it effectively does when we move from quantum to classical phenomena) the errors become swamped by our sloppy measurements and classical logic is a good approximation for practical purposes.
That’s really getting into a much deeper metaphysical question, and one that isn’t settled ground: whether the mathematical notions of the continuum and limits “obtain” in the real world. You assert, basically, that there’s no such thing as a limit since the approach can never be completed in a finite number of steps. Mind you, you’re in excellent company, but it’s a far deeper question than the OP remotely dealt with.
Careful, there are two different senses of “converge” going on here. Which do you really mean? “Converge” as in a limiting process (as Planck -> 0, quantum -> classical) or as in coming together (foot + floor = footfloor)?
So should you. I implied nothing of the sort. I asked about how we could discard the notion of an epistemological barrier and infer its ontology. Even if there’s no way, even potentially, for us to assign the exact values, how do we know that’s because that’s how things are, rather than because of the nature of our means to discover knowledge?
Just a follow up to see whether there’s an answer to this:
Is one macro object distinguished from another by electromagnetic force? In other words, is the essential difference between my feet and the floor under them the fact that the forces binding together the atoms in my feet do not interact with the forces that are binding together the atoms in the floor? Why don’t my feet [merge] with the floor and become one object whenever the outermost layer of atoms for each come into contact?
You could say that, I guess - or at least, we distinguish objects from each other like this, even they are “really” conglomerations of Avogadros (~10[sup]23[/sup]) of elementary particles (some of which do suddenly appear where we don’t classically expect them to.)
They all interact with each other - it is merely the strength of those forces which ultimately give rise to the distinctions we macroscopic objects make. The stuff of your feet is bound by electromagnetic bonds (of varioustypes). Since the strength of these forces drops of quickly (with the square of distance), any attractive electromagnetic forces from the floorstuff are not significant (and usually cancelled out, unless it’s become charged from people walking on it with balloon-shoes). When that other fundamental force, gravity, pulls your feet and the floor into contact, Avogadros of feet atoms meet Avogadros of floor atoms, and gravity pushes many of them together so tightly that their pettycoat of electrons is moved aside and they show each other their positve nucleus. This repulsion between the nuclei provides the reactive force against gravity which stops you sinking through the floor.
They are merged when in contact, in a manner of speaking. Following from above, some materials are such that, upon contact, lots and lots of atoms do form attractive electromagnetic bonds with the floor atoms which are as strong as their bonds with each other - the material is adhesive. Your feet are not such a material - any bonds they form with the floor are far weaker than those with each other. When you lift your feet to counteract the gravitational force, the electromagnetic bonds are not strong enough to resist you (although their ridges can help you form helpful bonds on a slope, say, and the feet of other creatures eg. geckos can provide bonds strong enough to counteract gravity altogether).
